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On measurability properties connected with the superposition operator. (English) Zbl 1070.28001

Let \(F\) be a class of real-valued functions defined on some set \(E\), measurable with respect to a \(\sigma\)-algebra \(S\). A function \(\phi: E\times\mathbb{R}\to \mathbb{R}\) is called to be sup-measurable with respect to \(F\) if for each \(f\in F\) the function \(\phi_f\) defined by \(\phi_f(x)= \phi(x,f(x))\) is \(S\)-measurable. The main result of the paper is Theorem 1 describing the maximal family \(F'\supset F\) consisting also of \(S\)-measurable functions such that \(\phi\) is sup-measurable with respect to \(F'\). There are also numerous examples explaining the role of axioms of set theory in questions concerning sup-measurability.

MSC:

28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
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